# Some Fixed Point Theorems on Ordered Metric Spaces and Application

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## Abstract

We present some fixed point results for nondecreasing and weakly increasing operators in a partially ordered metric space using implicit relations. Also we give an existence theorem for common solution of two integral equations.

## Keywords

Continuous Function Integral Equation Point Theorem Existence Theorem Contractive Condition## 1. Introduction

Existence of fixed points in partially ordered sets has been considered recently in [1], and some generalizations of the result of [1] are given in [2, 3, 4, 5, 6]. Also, in [1] some applications to matrix equations are presented, in [3, 4] some applications to periodic boundary value problem and to some particular problems are, respectively, given. Later, in [6] O'Regan and Petruşel gave some existence results for Fredholm and Volterra type integral equations. In some of the above works, the fixed point results are given for nondecreasing mappings.

We can order the purposes of the paper as follows.

First, we give a slight generalization of some of the results of the above papers using an implicit relation in the following way.

In [1, 3], the authors used the following contractive condition in their result, there exists Open image in new window such that

Afterwards, in [2], the authors used the nonlinear contractive condition, that is,

where Open image in new window is anondecreasing function with Open image in new window for Open image in new window , instead of (1.1). Also in [2], the authors proved a fixed point theorem using generalized nonlinear contractive condition, that is,

for Open image in new window , where Open image in new window is as above. In the Section 3, we generalized the above contractive conditions using the implicit relation technique in such a way that

for Open image in new window , where Open image in new window is a function as given in Section 2. We can obtain various contractive conditions from (1.4). For example, if we choose

in (1.4), then, we have (1.3). Similarly we can have the contractive conditions in [7, 8, 9] from (1.4).

In some of the above mentioned theorems, the fixed point results are given for nondecreasing mappings. Also in these theorems the following condition is used:

In Section 4, we give some examples such that two weakly increasing mappings need not be nondecreasing. Therefore, we give a common fixed point theorem for two weakly increasing operators in partially ordered metric spaces using implicit relation technique. Also we did not use the condition (1.6) in this theorem. At the end, to see the applicability of our result, we give an existence theorem for common solution of two integral equations using a result of the Section 4.

## 2. Implicit Relation

Implicit relations on metric spaces have been used in many articles. See for examples, [10, 11, 12, 13, 14, 15].

Let Open image in new window denote the nonnegative real numbers, and let Open image in new window be the set of all continuous functions Open image in new window satisfying the following conditions:

Open image in new window is nonincreasing in variables Open image in new window ;

implies Open image in new window ;

Open image in new window , for all Open image in new window .

Example 2.1.

Open image in new window where Open image in new window , Open image in new window , Open image in new window .

Let Open image in new window and Open image in new window . If Open image in new window then Open image in new window which implies Open image in new window , a contradiction. Thus Open image in new window and Open image in new window . Similarly, let Open image in new window and Open image in new window then Open image in new window If Open image in new window , then Open image in new window Thus Open image in new window is satisfied with Open image in new window . Also Open image in new window , for all Open image in new window . Therefore, Open image in new window .

Example 2.2.

Open image in new window , where Open image in new window .

Let Open image in new window and Open image in new window . If Open image in new window then Open image in new window which is a contradiction. Thus Open image in new window and Open image in new window Similarly, let Open image in new window and Open image in new window then we have Open image in new window If Open image in new window then Open image in new window Thus Open image in new window is satisfied with Open image in new window Also Open image in new window for all Open image in new window Therefore, Open image in new window .

Example 2.3.

Open image in new window where Open image in new window is right continuous and Open image in new window , Open image in new window for Open image in new window

Let Open image in new window and Open image in new window If Open image in new window then Open image in new window which is a contradiction. Thus Open image in new window and Open image in new window Similarly, let Open image in new window and Open image in new window then we have Open image in new window If Open image in new window , then Open image in new window Thus Open image in new window is satisfied with Open image in new window Also Open image in new window , for all Open image in new window Therefore, Open image in new window .

Example 2.4.

Open image in new window , where Open image in new window , Open image in new window , Open image in new window and Open image in new window

Let Open image in new window and Open image in new window Then Open image in new window Similarly, let Open image in new window and Open image in new window then we have Open image in new window If Open image in new window then Open image in new window Thus Open image in new window is satisfied with Open image in new window . Also Open image in new window , for all Open image in new window . Therefore, Open image in new window .

## 3. Fixed Point Theorem for Nondecreasing Mappings

We need the following lemma for the proof of our theorems.

Lemma 3.1 (See [16]).

Let Open image in new window be a right continuous function such that Open image in new window for every Open image in new window , then Open image in new window , where Open image in new window denotes the Open image in new window -times repeated composition of Open image in new window with itself.

Theorem 3.2.

hold. If there exists an Open image in new window with Open image in new window , then Open image in new window has a fixed point.

Proof.

From Open image in new window , we have Open image in new window . Therefore, letting Open image in new window in (3.16), we have Open image in new window This is a contradiction since Open image in new window for Open image in new window Thus Open image in new window is a Cauchy sequence in Open image in new window so there exists an Open image in new window with Open image in new window .

which is a contradiction to Open image in new window . Thus Open image in new window and so Open image in new window .

Remark 3.3.

Note that if we take that

implies Open image in new window ,

Instead of Open image in new window in Theorem 3.2, again we can have the same result.

If we combine Theorem 3.2 with Example 2.1, we obtain the following result.

Corollary 3.4.

hold. If there exists an Open image in new window with Open image in new window , then Open image in new window has a fixed point.

Remark 3.5.

Theorem Open image in new window of [2] follows from Example 2.3, Remark 3.3, and Theorem 3.2.

Remark 3.6.

We can have some new results from other examples and Theorem 3.2.

Remark 3.7.

and hence Open image in new window has a unique fixed point. If condition (3.27) fails, it is possible to find examples of functions Open image in new window with more than one fixed point. There exist some examples to illustrate this fact in [3].

## 4. Fixed Point Theorem for Weakly Increasing Mappings

Now we give a fixed point theorem for two weakly increasing mappings in ordered metric spaces using an implicit relation. Before this, we will define an implicit relation for the contractive condition of the theorem.

Let Open image in new window be the set of all continuous functions Open image in new window satisfying Open image in new window and the following conditions:

implies Open image in new window ;

Open image in new window and Open image in new window , for all Open image in new window .

We can easily show that, all functions in the Examples in Section 2 are in Open image in new window .

Definition 4.1 (See [17, 18]).

Let Open image in new window be a partially ordered set. Two mappings Open image in new window are said to be weakly increasing if Open image in new window and Open image in new window for all Open image in new window .

Note that, two weakly increasing mappings need not be nondecreasing.

Example 4.2.

then it is obvious that Open image in new window and Open image in new window for all Open image in new window . Thus Open image in new window and Open image in new window are weakly increasing mappings. Note that both Open image in new window and Open image in new window are not nondecreasing.

Example 4.3.

Let Open image in new window be endowed with the coordinate ordering, that is, Open image in new window and Open image in new window . Let Open image in new window be defined by Open image in new window and Open image in new window , then Open image in new window and Open image in new window . Thus Open image in new window and Open image in new window are weakly increasing mappings.

Example 4.4.

Thus Open image in new window and Open image in new window are weakly increasing mappings. Note that Open image in new window but Open image in new window , then Open image in new window is not nondecreasing. Similarly, Open image in new window is not nondecreasing.

Theorem 4.5.

hold, then Open image in new window and Open image in new window have a common fixed point.

Remark 4.6.

Note that, in this theorem we remove the condition "there exists an Open image in new window with Open image in new window '' of Theorem 3.2. Again we can consider the result of Remark 3.7 for this theorem.

Proof of Theorem 4.5.

From Open image in new window , we have Open image in new window . Therefore, letting Open image in new window in (4.32), we have Open image in new window . This is a contradiction since Open image in new window for Open image in new window . Thus Open image in new window is a Cauchy sequence in Open image in new window , so Open image in new window is a Cauchy sequence. Therefore, there exists an Open image in new window with Open image in new window .

which is a contradiction to Open image in new window . Thus Open image in new window and so Open image in new window .

Remark 4.7.

We can have some new results from Theorem 4.5 with some examples for Open image in new window .

For example, we can have the following corollary.

Corollary 4.8.

hold, then Open image in new window and Open image in new window have a common fixed point.

Proof.

Let Open image in new window , then it is obvious that Open image in new window . Therefore, the proof is complete from Theorem 4.5.

## 5. Application

Consider the integral equations

The purpose of this section is to give an existence theorem for common solution of (5.1) using Corollary 4.8. This section is related to those [19, 20, 21, 22].

Let Open image in new window be a partial order relation on Open image in new window .

Theorem 5.1.

Consider the integral equations (5.1).

(i) Open image in new window and Open image in new window Open image in new window are continuous;

for each Open image in new window and comparable Open image in new window ;

(iv) Open image in new window .

Then the integral equations (5.1) have a unique common solution Open image in new window in Open image in new window .

Proof.

Then Open image in new window is a partially ordered set. Also Open image in new window is a complete metric space. Moreover, for any increasing sequence Open image in new window in Open image in new window converging to Open image in new window , we have Open image in new window for any Open image in new window . Also for every Open image in new window , there exists Open image in new window which is comparable to Open image in new window and Open image in new window [6].

Hence Open image in new window for each comparable Open image in new window . Therefore, all conditions of Corollary 4.8 are satisfied. Thus the conclusion follows.

## Notes

### Acknowledgment

The authors thank the referees for their appreciation, valuable comments, and suggestions.

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